The Sombor index and reduced Sombor index, introduced by mathematical chemist Ivan Gutman [MATCH Commun. Math. Comput. Chem. 86 (2021) 11–16], are the recently proposed degree-based graph invariants that attained a lot of attention from researchers in a very short time. In this paper, the best possible upper bounds on the both aforementioned indices for molecular trees are obtained in terms of order and number of branching vertices or vertices of degree 2. The optimal molecular trees achieving the obtained bounds are also completely characterized.

The general $Z$-type index is a molecular descriptor, introduced recently by Chen and Lin [Discrete Optim., 50 (2023), 100808], which generalizes several well-known molecular descriptors, including the (general) sum-connectivity index and (general) Platt index. The primary objective of the current paper is to study the largest value of the general $Z$-type index of graphs in the class of all fixed-order trees (and chemical trees) with a particular number of segments.

This paper is concerned with a generalization of the atom-bond sum-connectivity (ABS) index, devised recently in [A. Ali, B. Furtula, I. Redžepović, I. Gutman, Atom-bond sum-connectivity index, J. Math. Chem., 60 (2022), 2081-2093]. For a connected graph $G$ with an order greater than $2$, the general atom-bond sum-connectivity index is represented as $AB{S}_{\gamma }(G)$ and is defined as the sum of the quantities $(1-2(d_x+d_y{)}^{-1}{)}^{\gamma }$ over all edges $xy$ of the graph $G$, where $d_x$ and $d_y$ represent the degrees of the vertices $x$ and $y$ of $G$, respectively, and $\gamma$ is any real number. For $-10\le \gamma \le 10$, the significance of $AB{S}_{\gamma }$ is examined on the data set of octane isomers for predicting six selected physicochemical properties of the mentioned compounds; promising results are obtained when the approximated value of $\gamma$ belongs to the set $\{-3,1,3\}$. The effect of the addition of an edge between two non-adjacent vertices of a graph under $AB{S}_{\gamma }$ is also investigated. Moreover, the graphs possessing the maximum value of $AB{S}_{\gamma }$, with $\gamma >0$, are characterized from the set of all connected graphs of a fixed order and a fixed (ⅰ) vertex connectivity not greater than a given number or (ⅱ) matching number.

This paper deals with the properties of the generalized Gutman–Milovanović index, generalized elliptic–Sombor index, generalized Zagreb–Sombor index, and general Euler–Sombor index. These include, as special cases, several previously studied molecular descriptors and most of their general versions; for instance, the general Randić index, the general sum-connectivity index, the general Sombor index, etc. The aforementioned descriptors are examined for their applicability in predicting 13 properties of octane isomers, and the results are compared with the ones generated by a benchmark data set (proposed by the International Academy of Mathematical Chemistry), containing 102 descriptors of octane isomers, and also with variable and discrete Adriatic indices. Although these descriptors slightly outperform the descriptors considered for comparison in several cases, a considerable improvement is detected in the case of boiling point. Several fundamental bounds and optimal results of the above-said descriptors are also reported.

Consider a graph $G$ having edge set $E$, and denote by $d_x$ the degree of a vertex $x$ in $G$. A unicyclic graph is defined as a connected graph containing exactly one cycle. This work focuses on unicyclic graphs of a fixed order and examines the graph invariants of such graphs of the form $BI{D}_{\varphi }(G)=\sum _{yz\in E}\varphi (d_y,d_z)$, where $\varphi$ is a symmetric and real-valued function. Such graph invariants are known as BID (bond incident degree) indices. The main objective is to determine the graphs that either minimize or maximize the quantity $BI{D}_{\varphi }$ among fixed-order unicyclic graphs with prescribed maximum degree, under specific assumptions on the function $\varphi$. These assumptions are satisfied by several classical and modern degree-based graph invariants. Generally, the results obtained are applicable to a wide range of such invariants. In particular, one of the obtained results covers the harmonic and sum-connectivity indices, while another applies to the recently proposed Sombor and Euler–Sombor indices as well as their reduced versions.

Let $G$ be a graph with edge set $E(G)$. Let $d_x$ denote the degree of a vertex $x$ in $G$. For a nonnegative integer $k$, a connected graph of order $n$ and size $n+k-1$ is called a $k$-cyclic graph. This paper is concerned with $k$-cyclic graphs and their graphical indices of the form $BI{D}_f(G)=\sum _{uv\in E(G)}f(d_u,d_v)$, where $f$ is a symmetric function whose outputs are real numbers. Particularly, the graphs minimizing or maximizing $BI{D}_f$ among all $k$-cyclic graphs with a given order are studied under certain constraints on $f$. Various existing indices meet these constraints, and hence the obtained results hold for those indices; more precisely, one of the obtained results covers the recently developed elliptic Sombor and Zagreb-Sombor indices, while another result covers the recently introduced Euler-Sombor index.

Consider a tree graph $G$ with edge set $E(G)$. The notation $d_G(x)$ represents the degree of vertex $x$ in $G$. Let $\mathfrak{f}$ be a symmetric real-valued function defined on the Cartesian square of the set of all distinct elements of the degree sequence of $G$. A graphical edge-weight-function index for the graph $G$, denoted by ${\mathcal{I}}_{\mathfrak{f}}(G)$, is defined as ${\mathcal{I}}_{\mathfrak{f}}(G)=\sum _{st\in E(G)}\mathfrak{f}(d_G(s),d_G(t))$. This paper establishes the best possible bounds for ${\mathcal{I}}_{\mathfrak{f}}(G)$ in terms of the order of $G$ and parameter $\mathfrak{p}$, subject to specific conditions on $\mathfrak{f}$. Here, $\mathfrak{p}$ can be one of the following three graph parameters: (ⅰ) matching number, (ⅱ) the count of pendent vertices, and (ⅲ) maximum degree. We also characterize all tree graphs that achieve these bounds. The constraints considered for $\mathfrak{f}$ are satisfied by several well-known indices. We specifically illustrate our findings by applying them to the recently introduced Euler-Sombor index.